Is there a stranger sports breed — or any breed, for that matter — on this planet than the head coach of an NFL team? He spends his weeks locked up in a sanitized practice facility, poring over game film, sleeping at his desk, dreaming of schemes that may give his team one extra yard. He’s a sweatshirted pope on a podium, trying to play mind games with the other team’s leader and invariably speaking to the assembled media in an indecipherable tongue. Hey, Belichick, are you saying your third wide receiver is healthy enough to play or not?

He presides over a sprawling bureaucracy. The offensive coordinator, defensive coordinator, wide-receivers coach, running-backs coach, linebackers coach, line coach, special-teams coach, clock-management guy and second assistant to the kick-holder coach all enjoy their little fiefdoms. Seriously, cutting the size of football-team governance would tack on another point to the unemployment rate.

As the NFL playoffs head into the divisional round this weekend, this bizarre breed is particularly intriguing. Start with the weekend’s marquee matchup, between the New York Jets and the New England Patriots. The last thing Americans need is more national focus on a New York–Boston sports rivalry; the hundreds of Yankees–Red Sox games televised every year are intolerable enough. But geography aside, Jets-Pats on Sunday, Jan. 16, will be a treat.

If you’re not already familiar with New York Jets coach Rex Ryan, here’s the snapshot: he’s a profane, trash-talking giant of a man who doesn’t seem to care what anyone thinks of him. That extends to sports fans and to the media speculation that he may have a foot fetish (Ryan hasn’t denied that his voice could be heard on a Web video, featuring a woman who looks like his wife, talking suggestively about the attractiveness of feet, but he has refused to discuss what he calls a “personal matter.”) Ryan is also a great coach who, in just two seasons, has won three playoff games with the Jets, one of the NFL’s most traditionally moribund franchises. Through it all, Ryan has displayed a fondness for psychological ploys, and the latest trick is to put all the focus on his babbling, thereby taking the pregame pressure off his players. Last week, he declared that New York’s game against the Indianapolis Colts was a “personal” affair between Colts quarterback Peyton Manning and him. Nothing could be further from the truth: his round behind wasn’t out there chasing the future Hall of Famer on the field. But Ryan’s bluster dominated the conversation, and his game plan helped the Jets shut down the high-powered Indianapolis offense during New York’s 17-16 upset.

So Ryan has raised his game even higher this week, declaring that the matchup between the Jets and New England is “about Bill Belichick vs. Rex Ryan.” The two men couldn’t be less alike. Whereas Ryan is gregarious, Belichick is dry at best. At worst he’s a cranky, socially awkward football menace; his critics claim he cheated his way to past Super Bowls by videotaping other teams’ signals, but few would deny that he has an unrivaled grasp of the complexities of pro football. In early December, New England destroyed the Jets 45-3, and Ryan said that during the postgame handshake Belichick “just looked” at him after the Jets coach said, “See you in Round 3.” (The Jets had beaten New England 28-14 in the second week of the season.) Such a reaction was pure Belichick: a little odd (why not just offer a friendly chuckle?) and probably calculated (the message being, “I can play a much quieter mind game, you loudmouthed fool”).

Ryan also took a shot at star New England quarterback Tom Brady’s work ethic, noting that Peyton Manning never would have skipped watching an opponent in order to go to a Broadway show, as Brady did during last Saturday’s Jets-Colts affair. As often happens in football, the players are now mimicking the personalities of their coaches. Jets defensive back Antonio Cromartie saw fit to curse out Brady, calling him an “a______” for his supposed taunting of a Jets sideline that deserved nothing but mockery for the team’s woeful effort in the 45-3 loss. Brady retorted with the type of dry wit that all of Belichick’s friends say the coach possesses but rarely displays in public, where he’s a snore. “Belichick has called me that. Our offensive coordinator has called me that,” Brady said. “I know that [the coaches] like me, so maybe [Cromartie] really likes me, because there are people who have called me that a few times.”

The mind games are swirling around Jet-Pats: on Thursday, Jan. 13, Patriots receiver Wes Welker made 10 cliché-laden references to feet during a press conference — “put your best foot forward,” “stick your toe in the water,” “have to be on your toes,” “being good little foot soldiers” — in what appeared to be a subliminal tweaking of Ryan. But the coaches offer subplots in all of the weekend’s contests. The other AFC matchup, between the Pittsburgh Steelers and the Baltimore Ravens, features Pittsburgh’s still baby-faced leader Mike Tomlin, who is seeking to make everyone under 40 feel guilty about their lack of accomplishments by winning a second Super Bowl in three years. On the other sideline is John Harbaugh, who is now officially fighting to get out of the shadow of his baby brother Jim, the Stanford coach whom many NFL teams fawned over after his team won the Orange Bowl. (Jim will now coach the San Francisco 49ers.) Early in the week, John Harbaugh jokingly said he was glad his Ravens broke Pittsburgh quarterback Ben Roethlisberger’s nose during an earlier Ravens-Steelers game. Roethlisberger, who attended the same college as Harbaugh (Miami University of Ohio), didn’t mind the poke, but other Steelers probably aren’t laughing. And in December, Pittsburgh wide receiver Hines Ward complained that Harbaugh offered him a “fake handshake” after a game. As if these two teams, division rivals known for their brutal defenses, needed any more reasons to beat each other up.

Over in the NFC, the Seattle Seahawks–Chicago Bears matchup offers another study in contrasts. Seattle has its jolly, rah-rah leader, Pete Carroll, the former USC coach who pushes his team to give it the ol’ college try. You can almost feel veteran quarterback Matt Hasselbeck rolling his eyes behind his facemask at his pep-rally leader, who many critics claim can’t succeed in the NFL with the school-spirit approach. (In fact, Carroll already didn’t succeed as a coach of both — you guessed it — the Jets and the Patriots in the 1990s.) Lovie Smith, the Bears’ coach, is part of the quieter, holier crowd led by former Colts coach Tony Dungy, one of his best friends. Smith never curses — so diligently so that I’m praying for a Jets-Bears Super Bowl, with Ryan egging Smith on, almost daring him to call him an a______.

Even the staid Mike-and-Mike pairing in Atlanta, in which the Falcons (and coach Mike Smith) will host the Packers (and Mike McCarthy), is underrated — perhaps in part because the matchup promises the most exciting offensive output of the weekend. Since taking over a 4-12 Falcons team in 2008, Smith has led Atlanta to the playoffs two of the past three seasons; the Falcons are the top seed in the conference, and “Smitty” is so energetic that, earlier in the season, he pulled a hamstring while running toward a ref to call a timeout. As for McCarthy, his decision to help nudge the aging hero Brett Favre out of Green Bay in 2008 is looking pretty smart. Favre’s replacement, Aaron Rodgers, oozes the kind of pocket confidence that has won Brady three Super Bowls.

As a general rule, both sports fans and sports media tend to ascribe too much importance to coaches. Do the NBA teams really need those dozen assistants sitting behind the bench? Couldn’t you or I lounge in a baseball dugout and decide whether the pitcher should bunt? But football is a much more complicated chess match, with the blitz packages, third-down packages and schemes with cryptic names like Cover 2. The sport’s one-game-a-week rhythm gives these baggy-eyed, obsessive mad scientists more time to shuffle X’s and O’s. In football, the coaches do matter. And the ones making the calls this weekend have plenty to prove. Especially the one who won’t shut up about his rival on the New England side of the field.